Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,..., β_n] is a Witt vector over k(x) = K₀, then the Witt equation y^p • y = β generates a tower of extensions through K_i = K_i-1(y_i) where y = [y₁, y₂,..., y_n]. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i-1(y_i); y_i^p - y_i = B_i, where, as a divisor in K_i-1, B_i has the form (B_i) = q Πp_j^λ_j. In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.